6.EE.9 8.F.1, 2, 3, and 4
Given a (linear or appropriate non-linear) equation, students should be able to make a connection between that algebraic equation (e.g. y=3x+4) and a graphical representation. Similarly, given a graphical representation, students should be able connect to and write a descriptive algebraic equation, assuming the graph is amenable to an algebraic description. This entry describes issues and strategies for helping students move easily between algebraic equations and graphical representations.
It is important for students to be able to construct a graph that represents an algebraic equation. Equally important is the ability to form an algebraic equation that represents a simple graph (such as a linear graph).
Research indicates that functions can be thought of as both a process and an object, and both types of conceptions are important for understanding graphing and equations of linear functions. The process conception of function regards functions as an input-output machine: if I plug in one value of x, I get out a different value of y. This conception is important because it allows students to use equations to plot points. In addition, this conception helps students to recognize that the graph of a line is actually a set of individual points, each of which are individual solutions which satisfy the algebraic equation. The object conception linear functions allows students to think of the entire function as a single object. As a single object, students can think about translating the line with vertical or horizontal translations and can identify holistic properties of the line such as “slope.” For example, a student with an object conception functions would be able to graph the equation y=2x+3 by taking the graph y=2x and shifting it vertically three units. See entries on Analysis of Change: Action/Process/Object conception of function.
Research indicates that students tend to think of changes in the “b” value of “y=mx+b” as a horizontal shift rather than a vertical shift. This intuition is not incorrect: lines of the same slope can be thought of as both horizontal and vertical shifts of one another. However, it is important for students to understand that a one unit increase in the value of “b” results in a vertical shift of one unit, whereas the same one unit shift in “b” results in a horizontal shift of -b/m units. (See entry on Analysis of Change: Shifting Linear Graphs).
In addition, research indicates that students have trouble with the fact that the standard linear equation “y=mx+b” does not explicitly list the value of the x-intercept. When writing equations for linear functions, students improperly place the value of the x-intercept into the standard equation of the line. (See entry on Analysis of Change: The Missing x-intercept Value in “y=mx+b”) Although teachers frequently disregard this as a trivial error, the relationship between the x-intercept and the values of “m” and “b” is important.
Consider the starburst pattern above. Using a graphing utility (such as grapher or a graphing calculator), recreate the starburst pattern.*
Mathematical Objective: The expectation is that preservice teachers should have the ability to recognize that the lines in the starburst pattern are of the form y=mx. However, students must think deeply about slope in order to have all of the lines evenly spaced. (For example, integer values for “m” do not result in the above figure.)
For formative assessment:
What are some of the possible values for the slope of the line that lies in the shaded region below? How do you know?*
a. Graph the equation y=x+4.
Jimmi said that this line would go through the axis at (4,0) because in the equation you add 4 to x. Do you think that Jimmi was right?
b. Graph the equation y = x and then change it to the equation y = x+3. Predict how the graph would change.**
Graphing calculators can be an effective tool for allowing students to quickly graph multiple functions on the same set of axis. By varying the values of “m” and “b” students can explore how the coefficients in the algebraic equation affect the graphical representation of the function. Graphing calculators provide linked approaches (numeric, graphical, and algebraic) to the same problem and allow complex math to become more accessible to a greater number of students.
Apple’s Grapher program (Included in the Utilities folder of the Apple OS X 10.4 and later) can also be an effective tool for this along with Web-based applications like Graph Sketcher by Shodor Interactivate (Use Apple Safari or Windows Internet Explorer for best results with this Java-based site).
To understand the relationship between algebraic expressions and their graphs, it is best to start with a simple case. Students should explore the case where m=1, and see the connection between how the changing value of “b” affects the x-intercept. Next, students should explore cases where the slope is not 1. As the value of the slope changes, students can see how the shift in “m” relates to the steepness of the slope of the graph, and a shift of the x-intercept toward the origin. This guides students away from seeing the x-intercept as a reflection of a change in “b” and seeing it as also depending on “m.”
Math Flyer is also an excellent app for learning the dynamics of the coefficients in equations on a graph.
Hennessy, S., Fung, P., & Scanlon, E. (2001). The role of the graphic calculator in mediating graphing activity. International Journal of Mathematical Education in Science and Technology, 32(2), 267-290.
*Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema & T. P.
Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.
**Moschkovich, J. (n.d.). "Students" use of the x-intercept as an instance of a transitional conception. Educational Studies in Mathematics, 37(2), 169.
Noble, T., Nemirovsky, R., DiMattia, C., & Wright , T. (2002). On Learning to See: How Do
Middle School Students Learn to Make Sense of Visual Representations in Mathematics?, TERC; Annual Meeting of the American Educational Research Association (AERA), 2002, New Orleans, LA.
Penglase, Marina and Arnold, Stephen. (1996). The Graphics Calculator in Mathematics Education: A Critical Review of Recent Research. Mathematics Education Research Journal, 1996, Vol8, No.1, 58-90
Representation, Vision, and Visualization. Duval, Raymond. 1999. Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?